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Theorem funcsetcestrclem5 16739
Description: Lemma 5 for funcsetcestrc 16744. (Contributed by AV, 27-Mar-2020.)
Hypotheses
Ref Expression
funcsetcestrc.s 𝑆 = (SetCat‘𝑈)
funcsetcestrc.c 𝐶 = (Base‘𝑆)
funcsetcestrc.f (𝜑𝐹 = (𝑥𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))
funcsetcestrc.u (𝜑𝑈 ∈ WUni)
funcsetcestrc.o (𝜑 → ω ∈ 𝑈)
funcsetcestrc.g (𝜑𝐺 = (𝑥𝐶, 𝑦𝐶 ↦ ( I ↾ (𝑦𝑚 𝑥))))
Assertion
Ref Expression
funcsetcestrclem5 ((𝜑 ∧ (𝑋𝐶𝑌𝐶)) → (𝑋𝐺𝑌) = ( I ↾ (𝑌𝑚 𝑋)))
Distinct variable groups:   𝑥,𝐶   𝑥,𝑋   𝜑,𝑥   𝑦,𝐶,𝑥   𝑦,𝑋   𝑥,𝑌,𝑦   𝜑,𝑦
Allowed substitution hints:   𝑆(𝑥,𝑦)   𝑈(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)

Proof of Theorem funcsetcestrclem5
StepHypRef Expression
1 funcsetcestrc.g . . 3 (𝜑𝐺 = (𝑥𝐶, 𝑦𝐶 ↦ ( I ↾ (𝑦𝑚 𝑥))))
21adantr 481 . 2 ((𝜑 ∧ (𝑋𝐶𝑌𝐶)) → 𝐺 = (𝑥𝐶, 𝑦𝐶 ↦ ( I ↾ (𝑦𝑚 𝑥))))
3 oveq12 6624 . . . . 5 ((𝑦 = 𝑌𝑥 = 𝑋) → (𝑦𝑚 𝑥) = (𝑌𝑚 𝑋))
43ancoms 469 . . . 4 ((𝑥 = 𝑋𝑦 = 𝑌) → (𝑦𝑚 𝑥) = (𝑌𝑚 𝑋))
54reseq2d 5366 . . 3 ((𝑥 = 𝑋𝑦 = 𝑌) → ( I ↾ (𝑦𝑚 𝑥)) = ( I ↾ (𝑌𝑚 𝑋)))
65adantl 482 . 2 (((𝜑 ∧ (𝑋𝐶𝑌𝐶)) ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → ( I ↾ (𝑦𝑚 𝑥)) = ( I ↾ (𝑌𝑚 𝑋)))
7 simprl 793 . 2 ((𝜑 ∧ (𝑋𝐶𝑌𝐶)) → 𝑋𝐶)
8 simprr 795 . 2 ((𝜑 ∧ (𝑋𝐶𝑌𝐶)) → 𝑌𝐶)
9 ovexd 6645 . . 3 ((𝜑 ∧ (𝑋𝐶𝑌𝐶)) → (𝑌𝑚 𝑋) ∈ V)
109resiexd 6445 . 2 ((𝜑 ∧ (𝑋𝐶𝑌𝐶)) → ( I ↾ (𝑌𝑚 𝑋)) ∈ V)
112, 6, 7, 8, 10ovmpt2d 6753 1 ((𝜑 ∧ (𝑋𝐶𝑌𝐶)) → (𝑋𝐺𝑌) = ( I ↾ (𝑌𝑚 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  Vcvv 3190  {csn 4155  cop 4161  cmpt 4683   I cid 4994  cres 5086  cfv 5857  (class class class)co 6615  cmpt2 6617  ωcom 7027  𝑚 cmap 7817  WUnicwun 9482  ndxcnx 15797  Basecbs 15800  SetCatcsetc 16665
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620
This theorem is referenced by:  funcsetcestrclem6  16740  funcsetcestrclem7  16741  funcsetcestrclem8  16742  funcsetcestrclem9  16743
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